Barbara knows that she will need to buy a new car in 4 years. The car will cost $15,000 by then. How much should she invest now at 10%, compounded quarterly, so that she will have enough to buy a new car? Round to the nearest cent.
Require that X*(1.025)^16 = 15,000.
X is the initial amount that you will need to invest.
The 1.025 is the factor by which balance increases by every 3 months. It does this 16 times in 4 years.
You can use logs to get the answer. I will use base 10 logs. (The base does not matter, as long as you are consistent and use the same log base on both sides of the equation)
log 15000/X = 16 log 1.025 = 0.1715818
15,000/X = 1.4845056
X = $10,104.37
To find out how much Barbara should invest now at 10% compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (amount of money Barbara will need in 4 years, which is $15,000).
P = the principal (amount of money she needs to invest now).
r = the annual interest rate (10% or 0.10).
n = the number of compounding periods per year (quarterly, so 4 quarters in a year).
t = the number of years (4 years).
Plug in the values into the formula:
$15,000 = P(1 + 0.10/4)^(4*4)
Simplify the equation:
$15,000 = P(1 + 0.025)^(16)
Next, solve for P by dividing both sides of the equation by (1 + 0.025)^(16):
P = $15,000 / (1 + 0.025)^(16)
Using a calculator, evaluate (1 + 0.025)^(16) ≈ 1.4573477.
P = $15,000 / 1.4573477
P ≈ $10,304.41
So, Barbara should invest approximately $10,304.41 now at 10% compounded quarterly in order to have enough money to buy a new car in 4 years.